Question

While taking a shower, you notice that the shower head is made up of 36 small round openings, each with a radius of 1.4 mm. You also determine that it takes 7.0 s for the shower to completely fill a 1- liter container you hold in the water stream. The water in your house is pumped by a pump in the basement, 6.2 m below the level of the shower head. The pump maintains an absolute pressure of 1.6 atmospheres. Neglect viscosity. Use g = 9.8 m/s/s, and 1 atmosphere = 101,300 Pa.

[1 point] (a) What is the total area of the openings in the shower head?

[2 points] (b) At what speed is the water emerging from the holes?

cconnected to the pump (assume it has a constant cross-sectional area)?

[2 points] (c) At what speed is the water flowing through the pipe

[2 points] (d) What is the radius of the pipe?

[1 point] (e) If you covered up six of the holes in the shower head with your thumb, with what speed would you expect the water to emerge from the remaining holes?

Answer 1

a)

n = number of small round openings = 36

r = radius of each small round opening = 1.4 mm = 1.4 x 10^-3 m

Total area of the openings in the shower head is given as

A_s = n r² = (36) (3.14) (1.4 x 10^-3)²

A_s = 2.22 x 10^-4 m²

b)

V = Volume flow rate = volume of container filled/time taken = 1 L/7 sec = 1.43 x 10^-4 m³/s

A_s = Total area of the openings in the shower head = 2.22 x 10^-4 m²

v_s = speed of water emerging from holes

Using Equation of continuity

V = A_s v_s

1.43 x 10^-4 = (2.22 x 10^-4) v_s

v_s = 0.64 m/s

c)

v_p = speed in pipe = ?

v_s = 0.64 m/s

P_p = pressure in pipe = 1.6 atm = 162120 Pa

P_s = pressure at the shower head = atmospheric pressure = 101325 Pa

h_s = height at shower head = 6.2 m

= density of water = 1000 kg/m³

Using Bernoulli's equation

P_p + (0.5) v_p² = P_s + (0.5) v_s² + g h_s

162120 + (0.5) (1000) v_p² = 101325 + (0.5) (1000) (0.64)² + (1000) (9.8) (6.2)

v_p = 0.58 m/s

d)

R = radius of the pipe

Using Equation of continuity

V = A_p v_p

1.43 x 10^-4 = (3.14) R² (0.58)

R = 8.86 x 10^-3 m

e)

n' = number of small round openings after covering six openings = 36 - 6 = 30

r = radius of each small round opening = 1.4 mm = 1.4 x 10^-3 m

Total area of the openings in the shower head is given as

A'_s = n r² = (30) (3.14) (1.4 x 10^-3)²

A'_s = 1.85 x 10^-4 m²

v'_s = speed of water emerging from holes

Using Equation of continuity

V = A'_s v'_s

1.43 x 10^-4 = (1.85 x 10^-4) v'_s

v'_s = 0.77 m/s